Why is it necessary for the moment of the forces on a rigid body in static equilibrium to be zero about any point, even if the position of that point is not on the body itself.

  • $\begingroup$ Because what do you think would happen if there were a point at which the net force is not zero? $\endgroup$ – Carl Witthoft Dec 7 '16 at 13:46
  • $\begingroup$ Succinctly put, of the body is in equilibrium then any point of reference can be used for calculating that equilibrium including being on the surface of the sun due to laws of motion #1... that body will remain at rest until its equilibrium its upset by another force acting upon it. $\endgroup$ – Rhodie Jan 7 '19 at 18:59

Consider a body to be in equilibrium under the action of forces $F_i$, $i=1,\ldots,n$ which act at locations $r_i$, $i=1,\ldots,n$.

Equilibrium of the body implies $$\sum _{i=1}^n r_i \times F_i=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum _{i=1}^n F_i=0$$

What about the moment about an arbitrary point p (that may or may not be on the body).

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That can be computed as

$$\sum _{i=1}^n (r_i-p) \times F_i = \sum _{i=1}^n r_i \times F_i-p\times\sum _{i=1}^n F_i = 0-p\times 0 = 0$$

Hence the result. The moment is zero about any point, whether it is on the body or not.

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